Chapter Eight

ELECTROMAGNETIC

WAVES

8.1 INTRODUCTION

In Chapter 4, we learnt that an electric current produces magnetic field

and that two current-carrying wires exert a magnetic force on each other.

Further, in Chapter 6, we have seen that a magnetic field changing with

time gives rise to an electric field. Is the converse also true? Does an

electric field changing with time give rise to a magnetic field? James Clerk

Maxwell (1831-1879), argued that this was indeed the case – not only

an electric current but also a time-varying electric field generates magnetic

field. While applying the Ampere’s circuital law to find magnetic field at a

point outside a capacitor connected to a time-varying current, Maxwell

noticed an inconsistency in the Ampere’s circuital law. He suggested the

existence of an additional current, called by him, the displacement

current to remove this inconsistency.

Maxwell formulated a set of equations involving electric and magnetic

fields, and their sources, the charge and current densities. These

equations are known as Maxwell’s equations. Together with the Lorentz

force formula (Chapter 4), they mathematically express all the basic laws

of electromagnetism.

The most important prediction to emerge from Maxwell’s equations

is the existence of electromagnetic waves, which are (coupled) time-

varying electric and magnetic fields that propagate in space. The speed

of the waves, according to these equations, turned out to be very close to

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the speed of light( 3 ×10

8

m/s), obtained from optical

measurements. This led to the remarkable conclusion

that light is an electromagnetic wave. Maxwell’s work

thus unified the domain of electricity, magnetism and

light. Hertz, in 1885, experimentally demonstrated the

existence of electromagnetic waves. Its technological use

by Marconi and others led in due course to the

revolution in communication that we are witnessing

today.

In this chapter, we first discuss the need for

displacement current and its consequences. Then we

present a descriptive account of electromagnetic waves.

The broad spectrum of electromagnetic waves,

stretching from γ rays (wavelength ~10

–12

m) to long

radio waves (wavelength ~10

6

m) is described. How the

electromagnetic waves are sent and received for

communication is discussed in Chapter 15.

8.2 DISPLACEMENT CURRENT

We have seen in Chapter 4 that an electrical current

produces a magnetic field around it. Maxwell showed

that for logical consistency, a changing electric field must

also produce a magnetic field. This effect is of great

importance because it explains the existence of radio

waves, gamma rays and visible light, as well as all other

forms of electromagnetic waves.

To see how a changing electric field gives rise to

a magnetic field, let us consider the process of

charging of a capacitor and apply Ampere’s circuital

law given by (Chapter 4)

“B

.

dl =

µ

0

i (t) (8.1)

to find magnetic field at a point outside the capacitor.

Figure 8.1(a) shows a parallel plate capacitor C which

is a part of circuit through which a time-dependent

current i (t) flows . Let us find the magnetic field at a

point such as P, in a region outside the parallel plate

capacitor. For this, we consider a plane circular loop of

radius r whose plane is perpendicular to the direction

of the current-carrying wire, and which is centred

symmetrically with respect to the wire [Fig. 8.1(a)]. From

symmetry, the magnetic field is directed along the

circumference of the circular loop and is the same in

magnitude at all points on the loop so that if B is the

magnitude of the field, the left side of Eq. (8.1) is B (2

π

r).

So we have

B (2πr) =

µ

0

i (t) (8 .2)

JAMES CLERK MAXWELL (1831–1879)

James Clerk Maxwell

(1831 – 1879) Born in

Edinburgh, Scotland,

was among the greatest

physicists of the

nineteenth century. He

derived the thermal

velocity distribution of

molecules in a gas and

was among the first to

obtain reliable

estimates of molecular

parameters from

measurable quantities

like viscosity, etc.

Maxwell’s greatest

acheivement was the

unification of the laws of

electricity and

magnetism (discovered

by Coulomb, Oersted,

Ampere and Faraday)

into a consistent set of

equations now called

Maxwell’s equations.

From these he arrived at

the most important

conclusion that light is

an electromagnetic

wave. Interestingly,

Maxwell did not agree

with the idea (strongly

suggested by the

Faraday’s laws of

electrolysis) that

electricity was

particulate in nature.

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Electromagnetic

Waves

Now, consider a different surface, which has the same boundary. This

is a pot like surface [Fig. 8.1(b)] which nowhere touches the current, but

has its bottom between the capacitor plates; its mouth is the circular

loop mentioned above. Another such surface is shaped like a tiffin box

(without the lid) [Fig. 8.1(c)]. On applying Ampere’s circuital law to such

surfaces with the same perimeter, we find that the left hand side of

Eq. (8.1) has not changed but the right hand side is zero and not

µ

0

i,

since no current passes through the surface of Fig. 8.1(b) and (c). So we

have a contradiction; calculated one way, there is a magnetic field at a

point P; calculated another way, the magnetic field at P is zero.

Since the contradiction arises from our use of Ampere’s circuital law,

this law must be missing something. The missing term must be such

that one gets the same magnetic field at point P, no matter what surface

is used.

We can actually guess the missing term by looking carefully at

Fig. 8.1(c). Is there anything passing through the surface S between the

plates of the capacitor? Yes, of course, the electric field! If the plates of the

capacitor have an area A, and a total charge Q, the magnitude of the

electric field E between the plates is (Q/A)/ε

0

(see Eq. 2.41). The field is

perpendicular to the surface S of Fig. 8.1(c). It has the same magnitude

over the area A of the capacitor plates, and vanishes outside it. So what

is the electric flux

Φ

E

through the surface S ? Using Gauss’s law, it is

E

0 0

1

= =

Q Q

A A

A

Φ

ε ε

=E

(8.3)

Now if the charge Q on the capacitor plates changes with time, there is a

current i = (dQ/dt), so that using Eq. (8.3), we have

d

d

d

d

d

d

Φ

E

t t

Q Q

t

=

=

ε ε

0 0

1

This implies that for consistency,

ε

0

d

d

Φ

E

t

= i (8.4)

This is the missing term in Ampere’s circuital law. If we generalise

this law by adding to the total current carried by conductors through

the surface, another term which is ε

0

times the rate of change of electric

flux through the same surface, the total has the same value of current i

for all surfaces. If this is done, there is no contradiction in the value of B

obtained anywhere using the generalised Ampere’s law. B at the point P

is non-zero no matter which surface is used for calculating it. B at a

point P outside the plates [Fig. 8.1(a)] is the same as at a point M just

inside, as it should be. The current carried by conductors due to flow of

charges is called conduction current. The current, given by Eq. (8.4), is a

new term, and is due to changing electric field (or electric displacement,

an old term still used sometimes). It is, therefore, called displacement

current or Maxwell’s displacement current. Figure 8.2 shows the electric

and magnetic fields inside the parallel plate capacitor discussed above.

The generalisation made by Maxwell then is the following. The source

of a magnetic field is not just the conduction electric current due to flowing

FIGURE 8.1 A

parallel plate

capacitor C, as part of

a circuit through

which a time

dependent current

i (t) flows, (a) a loop of

radius r, to determine

magnetic field at a

point P on the loop;

(b) a pot-shaped

surface passing

through the interior

between the capacitor

plates with the loop

shown in (a) as its

rim; (c) a tiffin-

shaped surface with

the circular loop as

its rim and a flat

circular bottom S

between the capacitor

plates. The arrows

show uniform electric

field between the

capacitor plates.

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charges, but also the time rate of change of electric field. More

precisely, the total current i is the sum of the conduction current

denoted by i

c

, and the displacement current denoted by i

d

(= ε

0

(d

Φ

E

/

dt)). So we have

0